Kostka polynomials and affine Kac-Moody algebras

PI建议研究仿射（以及更普遍的可对称）Kac-Moody代数的Hall-Littlewood和Kostka-Foulkes多项式的某些推广的表示理论和组合。具体来说，PI希望从Cherednik-Macdonald理论、q-超几何级数和正性的角度研究仿射Kostka-Foulkes多项式。PI还希望使用与任意Kac-Moody代数相对应的多项式来更多地阐明这些难以理解的李代数的表示。对称函数是多元多项式，它在变量置换时保持不变。他们承认这是一个涉及代数、组合和表示理论相互作用的丰富理论。Hall-Littlewood多项式是依赖于一个额外参数的对称函数，它在两类非常重要的对称函数之间进行插值。霍尔-利特尔伍德多项式理论的起源可以追溯到霍尔、利特尔伍德以及后来麦克唐纳的著作。这些多项式自然地出现在数学的各个领域，包括代数、几何、表示理论、组合学和数学物理。它们有许多重要的性质和应用。提出的研究目标是研究经典Hall-Littlewood多项式在无限维（Kac-Moody）李代数情况下的推广，并在这种一般情况下推导经典性质的类似物。PI一方面希望阐明Hall-Littlewood多项式的泛化与与之相关的经典对象（如“双仿射Hecke代数”）的泛化之间的关系。

The PI proposes to study the representation theory and combinatoricsunderlying certain generalizations of Hall-Littlewood and Kostka-Foulkes polynomials for affine (and more generally symmetrizable) Kac-Moody algebras. Specifically, the PI wishes to study affine Kostka-Foulkes polynomials from the perspectives of Cherednik-Macdonald theory, q-hypergeometric series and positivity. The PI also hopes to use the polynomials corresponding to arbitrary Kac-Moody algebras toshed more light on representations of these poorly understood Lie algebras.Symmetric functions are multivariate polynomials which remain unchangedwhen the variables are permuted. They admit a rich theory involving an interplay of algebra, combinatorics and representation theory. Hall-Littlewood polynomials are symmetric functions that depend on an extra parameter and interpolate between two very important classes of symmetric functions. The theory of Hall-Littlewood polynomials traces its origins to works of Hall, Littlewood and later of Macdonald. These polynomials occur naturally in diverse areas of mathematics includingalgebra, geometry, representation theory, combinatorics and mathematical physics. They have many important properties and applications.The goal of the proposed research is to study the generalization of the classical Hall-Littlewood polynomials to the case of infinite dimensional (Kac-Moody) Lie algebras and to derive analogs of classical properties in this general setting. The PI hopes to clarify the relation between the generalizations of Hall-Littlewood polynomials on one hand and the generalizations of the classical objects it is related to (such as the "double affine Hecke algebra").